Orthogonal and Orthonormal Vectors

This video tutorial introduces orthogonal and orthonormal sets of vectors, explaining their definitions and properties. It covers the standard basis vectors in R3 as examples of orthonormal sets and demonstrates how to check if a set of vectors is orthogonal using dot products. The tutorial also explains how to convert orthogonal sets into orthonormal sets by dividing vectors by their magnitudes.

What is the defining characteristic of an orthogonal set of vectors?

• All vectors are parallel.

• All vectors have the same magnitude.

• All vectors are unit vectors.

• All vectors are perpendicular to each other.

What additional property does an orthonormal set of vectors have compared to an orthogonal set?

• Vectors are unit vectors.

• Vectors have zero magnitude.

• Vectors are in the same direction.

• Vectors are parallel.

Which of the following is a standard basis vector in R3?

• (1, 0, 0)

• (2, 2, 2)

• (1, 1, 1)

• (0, 0, 0)

How do you determine if a set of two vectors is orthogonal?

• Check if their magnitudes are equal.

• Check if their dot product is zero.

• Check if their cross product is zero.

• Check if they are parallel.

What is the result of the dot product of a vector with itself if it is a unit vector?

• Its magnitude

• Negative one

• One

• Zero

In the example of a non-orthogonal set, what was the issue with the dot products?

• Vectors were parallel.

• All vectors were unit vectors.

• One dot product was not zero.

• All dot products were zero.

Why is it important for all dot products to be zero in an orthogonal set?

• To verify vectors are unit vectors.

• To confirm vectors are perpendicular.

• To check vectors have the same magnitude.

• To ensure vectors are parallel.

What is the first step in converting an orthogonal set to an orthonormal set?

• Subtract vectors from each other.

• Divide each vector by its magnitude.

• Multiply each vector by its magnitude.

• Add all vectors together.

What is the magnitude of a unit vector?

• Two

• Depends on the vector

• One

• Zero

What does dividing each component of a vector by its magnitude achieve?

• It makes the vector a unit vector.

• It makes the vector zero.

• It doubles the vector's length.

• It reverses the vector's direction.